Proposition III.4 — If in a circle two straight lines cut one another which are …

Proof

Let

A B

C D

be two chords intersecting at

E

, neither through the centre

F

. Suppose for contradiction that

E

bisects both:

A E = E B

and

C E = E D

. Join

F E

. By III.3 applied to chord

A B

(since

F

is the centre and

F E

bisects

A B

E

F E ⊥ A B

. Applied to

C D

, the same line

F E

⊥ C D

. But the perpendicular from

F

to a line is unique (I.11), so

A B

and

C D

must coincide — contradiction with their being two distinct chords.

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Depends on (3)

III.3Proposition III.3If in a circle a straight line through the centre bisect a straight line not through the centre, it also cuts it at…
I.11Proposition I.11To draw a straight line at right angles to a given straight line from a given point on it.
1Common notion 1Things which are equal to the same thing are also equal to one another.

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